1. Hany Ferdinando Dept. of Electrical Eng. Petra Christian University
Z Transform (1)
Hany Ferdinando
Dept. of Electrical Eng.
Petra Christian University

2. Z Transform (1) - Hany Ferdinando
Overview
Introduction
Basic calculation
RoC
Inverse Z Transform
Properties of Z transform
Exercise
Z Transform (1) - Hany Ferdinando

3. Z Transform (1) - Hany Ferdinando
Introduction
For discrete-time, we have not only Fourier analysis, but also Z transform
This is special for discrete-time only
The main idea is to transform signal/system from time-domain to z-domain  it means there is no time variable in the z-domain
Z Transform (1) - Hany Ferdinando

4. Z Transform (1) - Hany Ferdinando
Introduction
One important consequence of transform-domain description of LTI system is that the convolution operation in the time domain is converted to a multiplication operation in the transform-domain
Z Transform (1) - Hany Ferdinando

5. Z Transform (1) - Hany Ferdinando
Introduction
It simplifies the study of LTI system by:
Providing intuition that is not evident in the time-domain solution
Including initial conditions in the solution process automatically
Reducing the solution process of many problems to a simple table look up, much as one did for logarithm before the advent of hand calculators
Z Transform (1) - Hany Ferdinando

6. Z Transform (1) - Hany Ferdinando
Basic Calculation or
They are general formula:
Index ‘k’ or ‘n’ refer to time variable
If k > 0 then k is from 1 to infinity
Solve those equation with the geometrics series
Z Transform (1) - Hany Ferdinando

7. Z Transform (1) - Hany Ferdinando
Basic Calculation
Calculate:
Z Transform (1) - Hany Ferdinando

8. Z Transform (1) - Hany Ferdinando
Basic Calculation
Different signals can have the same transform in the z-domain  strange
The problem is when we got the representation in z-domain, how we can know the original signal in the time domain…
Z Transform (1) - Hany Ferdinando

9. Region of Convergence (RoC)
Geometrics series for infinite sum has special rule in order to solve it
This is the ratio between adjacent values
For those who forget this rule, please refer to geometrics series
Z Transform (1) - Hany Ferdinando

10. Region of Convergence (RoC)
Z Transform (1) - Hany Ferdinando

11. Region of Convergence (RoC)
Z Transform (1) - Hany Ferdinando

12. Region of Convergence (RoC)
Z Transform (1) - Hany Ferdinando

13. Z Transform (1) - Hany Ferdinando
RoC Properties
RoC of X(z) consists of a ring in the z-plane centered about the origin
RoC does not contain any poles
If x(n) is of finite duration then the RoC is the entire z-plane except possibly z = 0 and/or z = ∞
Z Transform (1) - Hany Ferdinando

14. Z Transform (1) - Hany Ferdinando
RoC Properties
If x(n) is right-sided sequence and if |z| = ro is in the RoC, then all finite values of z for which |z| > ro will also be in the RoC
If x(n) is left-sided sequence and if |z| = ro is in the RoC, then all values for which 0 < |z| < ro will also be in the RoC
Z Transform (1) - Hany Ferdinando

15. Z Transform (1) - Hany Ferdinando
RoC Properties
If x(n) is two-sided and if |z| = ro is in the RoC, then the RoC will consists of a ring in the z-plane which includes the |z| = ro
Z Transform (1) - Hany Ferdinando

16. Z Transform (1) - Hany Ferdinando
Inverse Z Transform
Use RoC
information
Direct division
Partial expansion
Alternative partial expansion
Z Transform (1) - Hany Ferdinando

17. Z Transform (1) - Hany Ferdinando
Direct Division
If the RoC is less than ‘a’, then expand it to positive power of z
a is divided by (–a+z)
If the RoC is greater than ‘a’, then expand it to negative power of z
a is divided by (z-a)
Z Transform (1) - Hany Ferdinando

18. Z Transform (1) - Hany Ferdinando
Partial Expansion
If the z is in the power of two or more, then use partial expansion to reduce its order
Then solve them with direct division
Z Transform (1) - Hany Ferdinando

19. Properties of Z Transform
General term and condition:
For every x(n) in time domain, there is X(z) in z domain with R as RoC
n is always from –∞ to ∞
Z Transform (1) - Hany Ferdinando

20. Z Transform (1) - Hany Ferdinando
Linearity
a x1(n) + b x2(n) ↔ a X1(z) + b X2(z)
RoC is R1∩R2
If a X1(z) + b X2(z) consist of all poles of X1(z) and X2(z) (there is no pole-zero cancellation), the RoC is exactly equal to the overlap of the individual RoC. Otherwise, it will be larger
anu(n) and anu(n-1) has the same RoC, i.e. |z|>|a|, but the RoC of [anu(n) – anu(n-1)] or d(n) is the entire z-plane
Z Transform (1) - Hany Ferdinando

21. Z Transform (1) - Hany Ferdinando
Time Shifting
x(n-m) ↔ z-mX(z)
RoC of z-mX(z) is R, except for the possible addition or deletion of the origin of infinity
For m>0, it introduces pole at z = 0 and the RoC may not include the origin
For m<0, it introduces zero at z = 0 and the RoC may include the origin
Z Transform (1) - Hany Ferdinando

22. Z Transform (1) - Hany Ferdinando
Frequency Shifting
ej(Wo)nx(n) ↔ X(ej(Wo)z)
RoC is R
The poles and zeros is rotated by the angle of Wo, therefore if X(z) has complex conjugate poles/zeros, they will have no symmetry at all
Z Transform (1) - Hany Ferdinando

23. Z Transform (1) - Hany Ferdinando
Time Reversal
x(-n) ↔ X(1/z)
RoC is 1/R
Z Transform (1) - Hany Ferdinando

24. Z Transform (1) - Hany Ferdinando
Convolution Property
x1(n)*x2(n) ↔ X1(z)X2(z)
RoC is R1∩R2
The behavior of RoC is similar to the linearity property
It says that when two polynomial or power series of X1(z) and X2(z) are multiplied, the coefficient of representing the product are convolution of the coefficient of X1(z) and X2(z)
Z Transform (1) - Hany Ferdinando

25. Z Transform (1) - Hany Ferdinando
Differentiation
RoC is R
One can use this property as a tool to simplify the problem, but the whole concept of z transform must be understood first…
Z Transform (1) - Hany Ferdinando

26. Z Transform (1) - Hany Ferdinando
Next…
For the next class, students have to read Z transform:
Signals and Systems by A. V. Oppeneim ch 10, or
Signals and Linear Systems by Robert A. Gabel ch 4, or
Sinyal & Sistem (terj) ch 10
Z Transform (1) - Hany Ferdinando